On Cosine Families Close to Scalar Cosine Families

A classic result, in its early form due to Cox [2], states that if A is a normed algebra with a unity denoted by 1 and a is an element of A such that supn∈N ‖a− 1‖ < 1, then a = 1. Cox’s version concerned the case of square matrices of a given size. This was later extended to bounded operators on Hilbert space by Nakamura and Yoshida [6], and to an arbitrary normed algebra by Hirschfeld [4] and Wallen [9]. The latter author proved in fact a stronger result, namely that ‖a − 1‖ = o(n) and lim infn→∞ n −1 (‖a− 1‖+ ‖a − 1‖+ · · ·+ ‖a − 1‖) < 1 imply a = 1, and he achieved this by using an elementary argument. An immediate consequence of the Cox–Nakamura–Yoshida–Hirschfeld–Wallen theorem is that if {S(t)}t≥0 is a semigroup on a Banach space X such that